Matrix factorizations and pentagon maps

We propose a specific class of matrices that participate in factorization problems that turn out to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang–Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order N = 2 matrices of this specific class are equivalent to the homogeneous normalization map . From order N = 3 matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang–Baxter maps.